Parallel combination of condenser

*“Parallel combination of condenser”:

in this combination first plate of all condenser are join at one point and second plate of each condenser is joint at another point. Some amount of charge is given at the first junction point and second is earthed.

In a figure shows the parallel combination of three condenser of capacitance C1, C2 and C3.  +Q charges given at the junction point A, this divides in three  part in the ratio of capacitance of condenser as Q1, Q2 and Q3 respectively and second junction point B is earthed. Hence two Plates of every condenser are joined between two fixed point as A and B. Thus the potential difference between their plates are same as (VA - VB).

we know that by the theory of condenser:
C = Q/∆V

Q = C×∆V

Charge on plates of 1st condenser:
Q1 = C1(VA - VB)

Charge on plates of 2nd condenser:
Q2 = C2(VA - VB)

Charge on plates of 1st condenser:
Q3 = C3(VA - VB)

Adding all above eqn

Q1 + Q2 + Q3 = C1(VA - VB)+C2(VA - VB)+
                            C3(VA - VB)

Q = (VA - VB) (C1 + C2 + C3 ) ------(1)

let the equivalent capacitance of all condenser are joined in parallel be C farad.
Thus all this condenser are replaced by a single condenser of capacitance C. the charge on its plates be Q and potential difference between its plates be (VA-VB).

Thus by the theory of condenser:
Charge on plates of equivalent condenser

Q = C (VA - VB) ------(2)

By comparing eqn1 and eqn2

C (VA - VB) = (VA - VB) (C1 + C2 + C3 )

[ C = C1 + C2 + C3 ]

This is the required expression for parallel combination of condenser.